= Normal form for free groups and free product of groups
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In a <free group>, a reduced word has no adjacent inverse letters. Successively cancelling such pairs gives the unique reduced representative in the chosen free generators. In a <free product>, a reduced word instead uses nonidentity factor elements as <syllables in a free product>, with consecutive <syllables in a free product> from different factors; the alternating <syllables in a free product> are unique.
For an <amalgamated free product> $A*_CB$, the usual reduced form relative to a common <subgroup> requires the maps $C\to A$ and $C\to B$ to be <injective>. Chosen representatives of <cosets> then distinguish the alternating factors and the remaining common-subgroup element. The exact form depends on the chosen left/right representative convention.
For general group <pushouts> defined by arbitrary <homomorphisms>, factors can collapse and this embedded-subgroup normal form is unavailable. For example, take $A=B=C=\mathbb Z$, map $C$ identically into $A$, and map it trivially into $B$. The pushout kills all of $A$ and is isomorphic to $B$, so words in nonidentity elements of $A$ cannot retain the usual nontrivial-factor normal form.
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